AMC 10 · 2002 · #4

Grade 6 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
order-of-operationspolynomial-factoring identify-subproblems ↑ Prerequisites: order-of-operations
📏 Short solution 💡 1 insight
Problem
The expression (3x-2)(4x+1) - (3x-2)·4x + 1 repeats the same factor (3x-2) in both of its products, and the final +1 stands outside both of them. Find the value of the expression when x=4.

Pick an answer.

(A)
$\ 0$
(B)
$\ 1$
(C)
$\ 10$
(D)
$\ 11$
(E)
$\ 12$

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Organize Information in More Ways

Both of the first two terms carry the same factor (3x-2), so instead of multiplying everything out, Tool #15 (Organize Information in More Ways) rewrites the expression to pull that shared factor to the front: (3x-2)[(4x+1)-4x]+1. The bracket collapses to a single number, which turns a messy problem into a one-line one. Tool #7 (Identify Subproblems) then handles the leftover in tidy steps — simplify the bracket, add the stray +1, substitute x=4. Tool #3 (Eliminate Possibilities) guards the built-in traps: dropping the outside +1 gives 10 (kills (C)), and mishandling the leftover constants -2+1 as 0 gives 12 (kills (E)).

1STEP 1

Pull out the shared factor

Both leading terms are (3x-2) times something, so factor it out: (3x-2)(4x+1)-(3x-2)·4x = (3x-2)[(4x+1)-4x]. The lone +1 rides along.

(3x-2)(4x+1)-(3x-2)·4x+1 = (3x-2)[(4x+1)-4x]+1
2STEP 2

Simplify the bracket

Inside the bracket, 4x and -4x cancel: (4x+1)-4x = 1. So the two products together are just (3x-2)·1 = 3x-2.

(4x+1)-4x = 1 → (3x-2)·1 = 3x-2
3STEP 3

Add the leftover +1

Bring back the +1 that waited outside: 3x-2+1. Combining constants -2+1 = -1 leaves the whole expression as 3x-1.

(3x-2)+1 = 3x-2+1 = 3x-1
4STEP 4

Substitute x=4

Substitute x=4 into 3x-1: 3·4-1 = 12-1 = 11, choice (D). Forgetting the outside +1 would give 10 in (C).

3·4-1 = 12-1 = 11 → (D)
Answer
11
Check the shortcut against a straight plug-in. At x=4: (3·4-2)=10 and (4·4+1)=17, so the first product is 10·17=170; the second term is 10·16=160; then 170-160+1=11. This matches the 3x-1=11 from the factoring shortcut, confirming (D). The answer also sits sensibly among the choices — it is not a wild number — and the near-misses 10 and 12 are exactly the values the two most common slips produce.
💡Key takeaway

When two terms share the same factor, pull it out front first — the mess often cancels and leaves a tiny expression to plug into.

  • Pull out the shared factor
  • Simplify the bracket
  • Add the leftover +1
  • Substitute x=4

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